Andrius Kulikauskas: I wish to show that my philosophy is very fruitful for developing a science of math, for understanding math as a cognitive language (implicit math) which takes place in the mind, and for overviewing all of math and how its branches and concepts unfold. Here are projects that I'm working on:

Understand the purpose of math and what distinguishes it from other languages and disciplines.

Abstraction?. Show how math unfolds, how its various branches and concepts arise, especially by studying the history of how math grows and documenting the ways of abstraction.

Describe and investigate the many dimensions of math, including discovery, beauty, insight, learning, humanity.

Math discovery Show how investigation can and does methodically apply the particular ways of figuring things out, notably, in mathematics, but also more generally.

Math connections Express my philosophy's concept in terms of mathematics and thus understand which mathematical concepts are most central.

Conversation?. Join with others for an ongoing conversation about collaborating on a science of math, especially, the "implicit math" that we think in our minds.

Ways of figuring things out in mathematics

Discovery. What are the ways of figuring things out in mathematics? We can study mathematics as an activity by which we create and solve mathematical problems. The techniques and structures that we use in our minds are much more elemental than the mathematical output which they generate.

Mathematical Concepts I am trying to organize an encyclopedia of mathematical concepts to see what they are and how they unfold.

Other questions about the big picture

Here are questions about the big picture in mathematics:

Beauty Mathematicians are guided by a sense of beauty. What is meant by beauty? What principles determine it? How does beauty lead to mathematical insight?

Education What resources are available to mathematicians that would help them most effectively learn mathematics so as to try to understand it as a whole? How might mathematicians collaborate effectively in trying to understand the big picture?

Insight What are the most fruitful insights in trying to understand mathematics? How can such insights best be stated?

Premathematics What concepts express intuitions that are prior to explicit mathematics and make it possible?

History? How can the history of mathematical discovery inform frameworks for the future development of mathematics?

Humanity? What parts or aspects of mathematics are specific to the human mind, body, culture, society, and what might be more broadly meaningful to other species in the universe?

How does Euler characteristic relate to homology, structures with holes?

What is the relationship between Pascal's triangle and the Grassmannian?

Questions I need to ask others

Why can't the field with one element be thought of as the zero ring?

Are my weights for the simplexes known?

Is my interpretation of the -1 simplex known?

Philosophical questions about math:

How is love (and life) related to duality, reflections, transformations and other math concepts?

How does 1 mediate the duality of 0 and infinity? And how is that duality variously broken?

What are the six basic transformation? and their relation with symmetry?

Math investigations:

List out the results of universal hyperbolic geometry and state them in terms of symmetric functions.

Write an elegant combinatorics of the finite field and interpret what is F1^n.

Consider more examples, simple and sophisticated, of how things are figured out in math.

What is the relationship between the surface math problem and the deep way of figuring things out?

How do we discover the right way to figure out a math problem?

How do we combine several distinct ways of figuring things out?

How can I apply my results to figure things out in math, the biggest problems?

Consider how infinity, zero and one are defined in the various geometries. How do these concepts fit together? How do they involve the viewer and their perspective? How might that relate to Christopher Alexander's principles of life and the plane of the viewer?

One way to think of geometry is in terms of what happens at infinity. For example, do all lines (through plane) meet at infinity at a common point? Or at a circle? Or do the ends of the lines not meet? Or do they go to a circle of infinite length?